# Linear Spaces (Vectors)

1. Jan 29, 2009

### kacete

1. The problem statement, all variables and given/known data
From the course of Linear Algebra and Analytic Geometry

I need to find the dimension and two different bases of subspace R3 generated by vectors (1,2,3), (4,5,6), (7,8,9).

2. Relevant equations
None.

3. The attempt at a solution
I tried

(a,b,c)=α1(1,2,3)+α2(4,5,6)+α3(7,8,9)

which became

a = α1 + 4α2 + 7α3
b = 2α1 + 5α2 + 8α3
c = 3α1 + 6α2 + 9α3

which (by Gaussian elimination) became an undetermined system with free variable α3.

4. The solution given by teacher
dim=2, example of bases={(1,0,-1),(0,1,2)} or {(2,1,0),(-1,0,1)}

I don't want the solution, I just want to understand the mechanics on how to find the bases and the generated subspace. If someone could explain it to me, thank you.

2. Jan 29, 2009

### Staff: Mentor

The system of equations you show would be used to find the span of your three vectors. Instead of equations that start with a= , b=, and c=, put 0 in for all three of those variables. You should end up with a row of zeroes and two nonzero rows.

What did you end up with when you row-reduced your matrix?

3. Jan 29, 2009

### kacete

So, I should have used the homogeneous system A . x = 0 ? Being A a matrix. Hmm...
I ended up with (after Gaussian elimination):

Code (Text):
[ 1  4  7 | a      ]
[ 0 -3 -6 | b-2a   ]
[ 0  0  0 | c-a-2b ]

4. Jan 29, 2009

### Staff: Mentor

If I can backpedal a bit, your work is fine. For the system represented by your augmented matrix to be consistent, it must be that c - a - 2b = 0.

or
Code (Text):

a = -2b + c
b =    b
c =          c

I added the 2nd and 3rd equations above so that I can get some vectors out of the equation c - a - 2b = 0. The equations I added are obviously true for all values of b and c, respectively.

Any vector [a b c]^T is a linear combination of [-2 1 0]^T and [1 0 1]^T. These come from setting b = 1, c = 0 and then b = 0, c = 1.

Different pairs of choices for b and c will give you different pairs of vectors for your basis.

Hope that helps.